I'm using the UpdateListItems operation to set a value in a date column. That works very well.

I also have a multi line column in the same list with the option "Append Changes to Existing Text" set.

When I use the UpdateListItems operation to set the date value, it seems to append the last text entered in the multi line column one more time so that it is duplicated. I think this only happend if I added new text to the multi line column in
the last version before I use the UpdateListItems.

This is the default behavior when dealing with a column that has "Append Changes to Existing Text". As a simple test, you can open the item up in edit view and then click OK or save. After you do that, SP will believe you have updated this item
and append the date to this field. The same behavior will happen whether you update the item through the GUI or update the item via Web Services.